EXAMPLE 2 Where are each of the following functions discontinuous? (a) f(x) x2 – 2x – 35 %3D x-7 if x + 0 (b) f(x) x5 1 if x = 0 %3D x2 2х- 35 if x + 7 (c) f(x) %3D х-7 if x = 7 (d) f(x) = [[ × ] %3D

Q: (c) There are four different values that are taken by the function f(x) = (100x) (x) List all of…

A: We recall some basics facts about Euler's function to solve this problem.

Q: 1 x - 5 Let f(x)= Then (fog)(5) (fog)(x) = ▪▪▪▪ and g(x) = 3x + 9.

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Q: x + 8 If f(x) = 7x – 8 and g(x) 7 (a) f(g(x)) = (b) g(f(x)) = (c) Thus g(x) is called an function of…

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Q: Evaluate the expression a+b a+b when a=23a=23 and b=45b=45. Write in simplest form

A: # the expressions are printed twice , taking them once given a=23 and b=45 The value of a+b is as…

Q: Which of the following functions is not one-to-one? (a) f(x) = x³ on (-00, 00) (b) f(x) = x² on (-0,…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: The one-to-one functions g and h are defined as follows. g={(-8, – 1), (-3, 7), (1, 9), (9, –8)} | h…

A: A function f(x) is said to be one-to-one if for every value of x in domain there is a unique value…

Q: 5. Find the domain of the function. (a) f(x)=4-5x+x² (b) f(x)= 1 x² - 6x+5

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Q: T + 3 If f(x) = 4x – 3 and g(x) %3D - 4 (a) f(9(x)) = %3D (b) g(f(x)) = (c) Thus g(x) is called an…

A: fx=4x-3 & gx=x+34 a). fgx=fx+34=4x+34-3=x+3-3=x b). gfx=g4x-3=4x-3+34=4x4=x

Q: The one-to-one functions g and h are defined as follows. g={(-3, 2), (2, 0), (6, 1), (7, - 2)} h (x)…

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Q: The function S (2) = x over the domain (0, 00) is shown. 10

A: Inverse function is find by switching x into y and y into x , then solve for y .

Q: x + 7 If f(æ) = 5x – 7 and g(æ) 5 (a) f(g(x) : (b) g(f(x)) =

A: Solution

Q: If f(x)=[x+4]-7 which of the following functions would translate f(x)1 unit to the right and 3 units…

A: Translation on f(x) Function  Translation Function after translation f(x) a units right f(x-a)…

Q: then A(-2u - 20) = -2 ---B 5 and Au 3 Au = 2 4

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Q: Which of the following functions has all of the following features? One negative x-intercept…

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Q: The one-to-one functions g and h are defined as follows. g={(-8, -4), (-3, 6), (1, 9), (2, 1)} h(x)…

A: Given, h(x)=3x+13 and h-1(x)=x-133

Q: i don't understand?

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Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.